Dehn invariant

Builds on Polyhedron Dihedral angle Tensor product of modules Abelian group

Definition

Let $P$ be a polyhedron in $\R^3$ . For each edge $e$ of $P$ , let $\ell(e)$ be its length and let $\theta(e)$ be the dihedral angle of $P$ at $e$ . The Dehn invariant of $P$ is the element $D(P) = \sum_{e} \ell(e) \otimes \theta(e)$ of the tensor product $\R \otimes_\Z (\R / \pi\Z)$ of abelian groups. The sum runs over all edges of $P$ .

Remarks

Example: for the cube of side $a$ , every dihedral angle is $\pi/2$ . So each edge gives $a \otimes \pi/2 = (a/2) \otimes \pi = 0$ , and $D(\text{cube}) = 0$ . A polyhedron with nonzero invariant: the regular tetrahedron of side $a$ has dihedral angle $\arccos(1/3)$ , which is not a rational multiple of $\pi$ . Its invariant $6a \otimes \arccos(1/3)$ is not $0$ , since $\R \otimes_\Z (\R/\pi\Z)$ is isomorphic to the $\Q$ -vector space $\R \otimes_\Q (\R/\Q\pi)$ . The invariant is additive: if $P$ is cut into polyhedra $P_1, \dots, P_k$ , then $D(P) = D(P_1) + \cdots + D(P_k)$ .

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Abelian group concept
Dihedral angle concept
Polyhedron concept
Tensor product of modules concept
Dehn invariant definition

Tag	`EYMHCPTZ8T`
URI	`m2://EYMHCPTZ8T`
Kind	definition
Writing	W0 stub
MSC	52B45, 51M20
Defines	Dehn invariant
Uses	Polyhedron, Dihedral angle, Tensor product of modules, Abelian group

Dehn invariant

Definition

Remarks

Reading path

Further reading

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